v2007.09.13 - Convex Optimization

202 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONSNext we use the Schur complement [203,6.4.3] [180] and matrixvectorization (2.2):minimizeX∈ S M , t∈Rsubject tot[tI vec(A − X)vec(A − X) T 1]≽ 0(485)S T XS ≽ 0This semidefinite program is an epigraph form in disguise, equivalentto (483); it demonstrates how a quadratic objective or constraint can beconverted to a semidefinite constraint.Were problem (483) instead equivalently expressed without the squareminimize ‖A − X‖ FX∈ S Msubject to S T XS ≽ 0(486)then we get a subtle variation:minimize tX∈ S M , t∈Rsubject to ‖A − X‖ F ≤ tS T XS ≽ 0(487)that leads to an equivalent for (486)minimizeX∈ S M , t∈Rsubject tot[tI vec(A − X)vec(A − X) T t]≽ 0(488)S T XS ≽ 03.1.7.2.1 Example. Schur anomaly.Consider a problem abstract in the convex constraint, given symmetricmatrix Aminimize ‖X‖ 2X∈ S M F − ‖A − X‖2 F(489)subject to X ∈ Cthe minimization of a difference of two quadratic functions each convex inmatrix X . Observe equality